MILNOR FIBRATIONS OF MEROMORPHIC FUNCTIONS

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Niveau: Supérieur, Doctorat, Bac+8
MILNOR FIBRATIONS OF MEROMORPHIC FUNCTIONS ARNAUD BODIN, ANNE PICHON, JOSE SEADE Abstract. In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punc- tured discs around the critical values of f/g, and the Milnor fibra- tion on a sphere. 1. Introduction The classical fibration theorem of Milnor in [6] says that every holo- morphic map (germ) f : (Cn, 0) ? (C, 0) with n > 2 and a critical point at 0 ? Cn has two naturally associated fibre bundles, and both of these are equivalent. The first is: (1) ? = f |f | : S? \K ?? S1 where S? is a sufficiently small sphere around 0 ? Cn and K = f?1(0)? S? is the link of f at 0. The second fibration is: (2) f : B? ? f?1(∂D?) ?? ∂D? ?= S1 where B? is the closed ball in Cn with boundary S? and D? is a disc around 0 ? C which is sufficiently small with respect to ?. The set N(?, ?) = B??f?1(∂D?) is usually called a local Milnor tube for f at 0, and it is diffeomorphic to S? minus an open regular neigh- bourhood T of K.

• truncated global


• milnor's proof concerns


• meromorphic function


• see also


• local milnor


• global milnor


• milnor


• lf ?

Sujets

• Meromorphic function
• See also

Informations

• Publié par: mijec
• Nombre de lectures: 24
• Langue: English

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MILNORFIBRATIONSOFMEROMORPHIC FUNCTIONS ARNAUDBODIN,ANNEPICHON,JOSE ´ SEADE Abstract. Inanalogywiththeholomorphiccase,wecomparethe topologyofMilnorfibrationsassociatedtoameromorphicgerm f/g:thelocalMilnorfibrationsgivenonMilnortubesoverpunc- tureddiscsaroundthecriticalvaluesoff/g,andtheMilnorfibra- tiononasphere. 1. Introduction TheclassicalfibrationtheoremofMilnorin[6]saysthateveryholo- morphicmap(germ)f:( C n ,0) → ( C ,0)withn>2andacritical pointat0 ∈ C n hastwonaturallyassociatedfibrebundles,andboth oftheseareequivalent.Thefirstis: f (1)φ=: S ε \ K −→ S 1 | f | where S ε isasufficientlysmallspherearound0 ∈ C n andK=f − 1 (0) ∩ S ε isthelinkoffat0.Thesecondfibrationis: (2)f: B ε ∩ f − 1 (∂ D δ ) −→ ∂ D δ = ∼ S 1 where B ε istheclosedballin C n withboundary S ε and D δ isadisc around0 ∈ C whichissufficientlysmallwithrespecttoε. ThesetN(ε,δ)= B ε ∩ f − 1 (∂ D δ )isusuallycalledalocalMilnortube forfat0,anditisdiffeomorphicto S ε minusanopenregularneigh- bourhoodTofK.(Thus,togettheequivalenceofthetwofibrations onehasto“extend”thelatterfibrationtoT \ K.)Infact,inorderto havethesecondfibrationoneneedstoknowthateverymap-germfas abovehastheso-called“Thomproperty”,whichwasnotknownwhen Milnorwrotehisbook.Whatheprovesisthatthefibersin(1)are diffeomorphictotheintersectionf − 1 (t) ∩ B ε fortcloseenoughto0. Thestatementthat(2)isafibrebundlewasprovedlaterin[5]byLeˆ Date:February16,2009. 2000MathematicsSubjectClassification.14J17,32S25,57M25. Keywordsandphrases.Meromorphicfunctions,Milnorfibration,Semitame maps,(i)-tame. 1

2ARNAUDBODIN,ANNEPICHON,JOSE ´ SEADE Du˜ngTra´nginthemoregeneralsettingofholomorphicmapsdefined onarbitrarycomplexanalyticspaces,andwecallittheMilnor-Leˆfi- brationoff.Onceweknowthat(2)isafibrebundle,thearguments of[6,Chapter5]showthisisequivalenttotheMilnorfibration(1). Theliteratureaboutthesefibrationsisvast,andsoaretheirgen- eralizationstovarioussettings,includingrealanalyticmap-germsand meromorphicmaps,andthatisthestartingpointofthisarticle. LetUbeanopenneighbourhoodof0in C n andletf,g:U −→ C be twoholomorphicfunctionswithoutcommonfactorssuchthatf(0)= g(0)=0.LetusconsiderthemeromorphicfunctionF=f/g:U → C P 1 definedby(f/g)(x)=[f(x)/g(x)].Asin[3],twosuchgermsat0, F=f/gandF 0 =f 0 /g 0 areconsideredasequal(orequivalent)ifand onlyiff=hf 0 andg=hg 0 forsomeholomorphicgermh: C n → C suchthath(0) 6 =0.Noticethatf/gisnotdefinedonthewholeU;its indeterminationlocusis  I=z ∈ U | f(x)=0andg(x)=0. Inparticular,thefibersofF=f/gdonotcontainanypointofI:for eachc ∈ C ,thefiberF − 1 (c)istheset  F − 1 (c)=x ∈ U | f(x) − cg(x)=0 \ I. Inaseriesofarticles,S.M.Gusein-Zade,I.LuengoandA.Melle- Herna´ndez,andlaterD.SiersmaandM.Tibaˇr,studiedlocalMilnor fibrationsofthetype(2)associatedtoeverycriticalvalueofthemero- morphicmapF=f/g.Seeforinstance[3,4],orTibar’sbook[12]and thereferencesinit.Ofcoursethe“Milnortubes” B ε ∩ F − 1 (∂ D δ )in thiscasearenotactualtubesingeneral,sincetheymaycontain0 ∈ U intheirclosure.Theseareinfact“pinchedtubes”. Itisthusnaturaltoaskwhetheronehasformeromorphicmap- germsfibrationsofMilnortype(1),andifso,howthesearerelatedto thoseoftheMilnor-Leˆtype(2)studied(forinstance)in[3,4,12].The firstofthesequestionswasaddressedin[10,1,11]fromtwodifferent viewpoints,whiletheanswertothesecondquestionisthebulkofthis article. Infact,itisprovedin[1]thatifthemeromorphicgermF=f/gis semitame(seethedefinitioninSection2),then g/fF 1 (3) | F | = | f/g | : S ε \ (L f ∪ L g ) −→ S isafiberbundle,whereL f = { f=0 }∩ S ε andL g = { g=0 }∩ S ε are theorientedlinksoffandg.NoticethatawayfromthelinkL f ∪ L g